#7184800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,6)(3,5,7,8)(10,14)(12,15)(17,18)(19,29)(20,26)(21,35)(22,27)(23,38)(24,28)(25,32)(30,46)(31,40)(33,42)(34,44)(36,41)(37,45)(39,43)(47,48);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,12,16,15)(17,19,24,33)(18,25,28,43)(20,30,21,36)(22,32,23,39)(26,40,35,47)(27,42,38,29)(31,34,48,37)(41,44,46,45);(2,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,24,21)(18,26,28,35)(19,30,33,36)(22,37,23,34)(25,40,43,47)(27,45,38,44)(29,46,42,41)(31,39,48,32);(10,14)(11,16)(17,21,24,20)(18,26,28,35)(19,31,33,48)(22,37,23,34)(25,41,43,46)(27,44,38,45)(29,47,42,40)(30,39,36,32);(9,12,13,15)(10,11,14,16)(17,22,24,23)(18,27,28,38)(19,32,33,39)(20,34,21,37)(25,42,43,29)(26,44,35,45)(30,48,36,31)(40,46,47,41)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7091996);